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ε

_{0}- L dI/dt = RI

The solution to this differential equation will be an exponential function going asymptotically towards the final current ε

_{0}/R. I have some trouble understanding the physical reason for this asymptotic behaviour. Intuitively it tells me that in the beginning the current rises a lot due to no backemf, but then the backemf grows and grows and grows until in the end where it is almost equal to that produced by the battery. Is this correct and come someone explain, in some detail, what kind of behaviour this differential equation describes? For instance, I can't see why the back emf should get bigger and bigger.